Skein Spaces and Spin Structures

This paper relates skein spaces based on the Kauffman bracket and spin structures. A spin structure on an oriented 3-manifold provides an isomorphism between the skein space for parameter A and the skein space for parameter −A. There is an application to Penrose’s binor calculus, which is related to the tensor calculus of representations of SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane, and the matrices are determined by a type of spinor transport which generalises to links in any 3-manifold. A second application shows that there is a skein space which is the algebra of functions on the set of spin structures for the 3-manifold. This paper relates skein spaces based on the Kauffman bracket and spin structures. The main result for a general parameter A is that a spin structure on an oriented 3-manifold provides an isomorphism between the skein spaces for parameters ±A. Specialising to the case of A = ±1 gives the application to Penrose’s binor calculus. The binor calculus is related to a tensor calculus of invariants for the group SU(2). The perspective developed here is that this tensor calculus is actually a calculus of spinors on the plane, and the matrices are determined by a type of spinor transport which generalises to links in any 3-manifold. As an elementary example, the unknot corresponds to the operation of transporting a spinor in C one full turn around a circle. This operation on spin space is minus the identity in SU(2), which has trace −2, the Kauffman bracket evaluation for the unknot for parameter A = ±1. However, the binor calculus is related to A = −1, whereas the geometrical description in terms of spinors occurs for A = 1. The isomorphism of the two skein spaces provides the relation between these. More generally, the skein spaces for A = 1 have a quotient which is a commutative algebra. For A = −1 this is known to be related to the algebra of functions on the space of flat SU(2)-connections on M . The analogous description for A = 1 is given here in terms of the flat connections over the frame bundle, where the holonomy around the fibres of the frame bundle is non-trivial. In the case of a primitive cube root of 1, the commutative skein algebra is the algebra of functions on the set of spin structures of the manifold M . This result follows naturally by using the isomorphism with the skein space for −A, a primitive sixth root of 1, for which it is easily shown that the algebra is the algebra of functions on H(M,Z2). Typeset by AMS-TEX